On the Complexity of Immersed Normal Surfaces

TL;DR

This paper explores the computational complexity of a relaxed version of normal surfaces in 3-manifold theory, proving NP-hardness for immersion decision and providing efficient algorithms for local variants.

Contribution

It introduces a relaxed notion of normal surfaces without quadrilateral conditions and establishes NP-hardness for immersion testing, along with polynomial algorithms for local cases.

Findings

Deciding immersion of relaxed normal surfaces is NP-hard.

A polynomial-time algorithm exists for a local version of the problem.

The proof uses a reduction from Boolean constraint satisfaction problems.

Abstract

Normal surface theory, a tool to represent surfaces in a triangulated 3-manifold combinatorially, is ubiquitous in computational 3-manifold theory. In this paper, we investigate a relaxed notion of normal surfaces where we remove the quadrilateral conditions. This yields normal surfaces that are no longer embedded. We prove that it is NP-hard to decide whether such a surface is immersed. Our proof uses a reduction from Boolean constraint satisfaction problems where every variable appears in at most two clauses, using a classification theorem of Feder. We also investigate variants, and provide a polynomial-time algorithm to test for a local version of this problem.